Brief history of “programming”
• “Programming” in the sense of devising
rules, methods, procedures, recipes,
algorithms, has always been present in
human thought.
• “Algorithm” originates from the name of
Al-Khwarizmi, Arab mathematician from
800AD.
Al-Khwarizmi, 860AD
“Algorists” 1504AD
Sumerian division algorithm,
2000BC
Sumerian division, translated
[The number is 4;10. What is its inverse?
Proceed as follows.
Form the inverse of 10, you will find 6.
Multiply 6 by 4, you will find 24.
Add on 1, you will find 25.
Form the inverse of 25], you will find 2;24.
[Multiply 2;24 by 6], you will find 14;24.
[The inverse is 14;24]. Such is the way to proceed.
Sumerian division, symbolically
The number is x. What is its inverse?
Proceed as follows.
Form the inverse of y, you will find y’.
Multiply y’ by z. You will find t.
Add on 1. You will find u.
Form the inverse of u. You will find u’ .
Multiply u’ by y’. You will find v.
The inverse is v. Such is the way to proceed.
Euclid’s algorithm for GCD
Proposition 2
Given two numbers not prime to one another, to find their greatest common
measure.
Let AB, CD be the two given numbers not prime to one another.
Thus it is required to find the greatest common measure of AB, CD.
…..
But, if CD does not measure AB, then the less of the numbers AB, CD, being
continually subtracted from the greater, some number will be left which will
measure the one before it.
…Let such a number measure them, and let it be G.
Now, since G measures CD, while CD measures BE, G also measures BE.
Functional vs imperative
• The Sumerian division algorithm, which is
representative of most algorithms in Mathematics,
is an example of a functional computation.
• The Euclid’s GCD algorithm is an example of an
imperative computation, involving iteration and
state change.
• [For more details, see: Jean-Luc Chabert: A
History of Algorithms, Springer, 1994]
Functional computation
• Apply functions to input
values or intermediate
values to compute new
outputs.
• Type checking helps
ensure the “plugcompatibility” of the
inputs/outputs.
Imperative computation
• Commands operate on the
store, which contain
locations, and read or
write them.
• One view is: commands
transform the state of the
store (i.e., “functions” of
some sort).
• Types are not of much
help to ensure correctness.
Store
Commands
Introduction to Haskell
(Borrowed from John Mitchell)
Language Evolution
Lisp
Algol 60
Algol 68
Pascal
C
Smalltalk
ML
Haskell
Modula
C++
Java
Many others: Algol 58, Algol W, Scheme, EL1, Mesa (PARC), Modula-2,
Oberon, Modula-3, Fortran, Ada, Perl, Python, Ruby, C#, Javascript, F#…
C Programming Language
Dennis Ritchie, ACM Turing Award for Unix
• Statically typed, general purpose systems programming
language
• Computational model reflects underlying machine
• Relationship between arrays and pointers
– An array is treated as a pointer to first element
– E1[E2] is equivalent to ptr dereference: *((E1)+(E2))
– Pointer arithmetic is not common in other languages
• Not statically type safe
– If variable has type float, no guarantee value is floating pt
• Ritchie quote
– “C is quirky, flawed, and a tremendous success”
ML programming language
• Statically typed, general-purpose programming language
– “Meta-Language” of the LCF theorem proving system
• Type safe, with formal semantics
• Compiled language, but intended for interactive use
• Combination of Lisp and Algol-like features
–
–
–
–
–
–
Expression-oriented
Higher-order functions
Garbage collection
Abstract data types
Module system
Exceptions
• Used in printed textbook as example language
Robin Milner, ACM Turing-Award for ML, LCF Theorem Prover, …
Haskell
• Haskell programming language is
– Similar to ML: general-purpose, strongly typed, higher-order,
functional, supports type inference, interactive and compiled use
– Different from ML: lazy evaluation, purely functional core, rapidly
evolving type system
• Designed by committee in 80’s and 90’s to unify research
efforts in lazy languages
– Haskell 1.0 in 1990, Haskell ‘98, Haskell’ ongoing
– “A History of Haskell: Being Lazy with Class” HOPL 3
Paul Hudak
John Hughes
Simon
Peyton Jones
Phil Wadler
Haskell B Curry
• Combinatory logic
– Influenced by Russell and Whitehead
– Developed combinators to represent
substitution
– Alternate form of lambda calculus that
has been used in implementation
structures
• Type inference
– Devised by Curry and Feys
– Extended by Hindley, Milner
Although “Currying” and “Curried functions”
are named after Curry, the idea was invented
by Schoenfinkel earlier
Why Study Haskell?
• Good vehicle for studying language concepts
• Types and type checking
–
–
–
–
General issues in static and dynamic typing
Type inference
Parametric polymorphism
Ad hoc polymorphism (aka, overloading)
• Control
– Lazy vs. eager evaluation
– Tail recursion and continuations
– Precise management of effects
Why Study Haskell?
• Functional programming will make you think
differently about programming.
– Mainstream languages are all about state
– Functional programming is all about values
• Haskell is “cutting edge”
– A lot of current research is done using Haskell
– Rise of multi-core, parallel programming likely to
make minimizing state much more important
• New ideas can help make you a better
programmer, in any language
Practitioners
Most Research Languages
1,000,000
10,000
Geeks
100
The quick death
1
1yr
5yr
10yr
15yr
Practitioners
Successful Research Languages
1,000,000
10,000
Geeks
100
The slow death
1
1yr
5yr
10yr
15yr
Practitioners
C++, Java, Perl, Ruby
Threshold of immortality
1,000,000
10,000
The complete
absence of death
Geeks
100
1
1yr
5yr
10yr
15yr
Practitioners
Committee languages
1,000,000
10,000
Geeks
100
The slow death
1
1yr
5yr
10yr
15yr
Practitioners
Haskell
1,000,000
10,000
“I'm already looking at coding
problems and my mental
perspective is now shifting
back and forth between purely
OO and more FP styled
solutions”
(blog Mar 2007)
Geeks
100
“Learning Haskell is a great way of
training yourself to think functionally so
you are ready to take full advantage of
C# 3.0 when it comes out”
(blog Apr 2007)
The second life?
1
1990
1995
2000
2005
2010
Function Types in Haskell
In Haskell, f :: A  B means for every x  A,
f(x) =
some element y = f(x)  B
run forever
In words, “if f(x) terminates, then f(x)  B.”
In ML, functions with type A  B can throw an
exception or have other effects, but not in Haskell
Basic Overview of Haskell
• Interactive Interpreter (ghci): read-eval-print
– ghci infers type before compiling or executing
– Type system does not allow casts or other loopholes!
• Examples
Prelude> (5+3)-2
6
it :: Integer
Prelude> if 5>3 then “Harry” else “Hermione”
“Harry”
it :: [Char]
-- String is equivalent to [Char]
Prelude> 5==4
False
it :: Bool
Overview by Type
• Booleans
True, False :: Bool
if … then … else …
--types must match
• Integers
0, 1, 2, … :: Integer
+, * , …
:: Integer
-> Integer -> Integer
• Strings
 “Ron
Weasley”
• Floats
1.0, 2, 3.14159, …
--type classes to disambiguate
Haskell Libraries
Simple Compound Types
 Tuples
(4, 5, “Griffendor”) :: (Integer, Integer, String)
 Lists
[] :: [a]
-- polymorphic type
1 : [2, 3, 4] :: [Integer]
-- infix cons notation
 Records
data Person = Person {firstName :: String,
lastName :: String}
hg = Person { firstName = “Hermione”,
lastName = “Granger”}
Patterns and Declarations
• Patterns can be used in place of variables
<pat> ::= <var> | <tuple> | <cons> | <record> …
• Value declarations
–General form:
–Examples
<pat> = <exp>
myTuple = (“Flitwick”, “Snape”)
(x,y) = myTuple
myList = [1, 2, 3, 4]
z:zs = myList
–Local declarations
• let (x,y) = (2, “Snape”) in x * 4
Functions and Pattern Matching
• Function declaration form
<name> <pat1> = <exp1>

<name> <pat2> = <exp2> …
<name> <patn> = <expn> …
• Examples
f(x,y) = x+y
--argument must match pattern (x,y)
length [] = 0
length (x:s) = 1 + length(s)
More Functions on Lists
• Append lists
–
append
([], ys) = ys
append
(x:xs, ys) = x : append (xs, ys)
–
• Reverse a list
–
reverse
[] = []
reverse
(x:xs) = (reverse xs) ++ [x]
–
• Questions
– How efficient is reverse?
– Can it be done with only one pass through list?
More Efficient Reverse
reverse xs =
let rev ( [], accum ) = accum
rev ( y:ys, accum ) = rev ( ys, y:accum )
in rev ( xs, [] )
1
3
2
2
3
3
1
3
2
2
1
1
List Comprehensions
• Notation for constructing new lists from old:
myData = [1,2,3,4,5,6,7]
twiceData = [2 * x | x <- myData]
-- [2,4,6,8,10,12,14]
twiceEvenData = [2 * x| x <- myData, x `mod` 2 == 0]
-- [4,8,12]
• Similar to “set comprehension”
{ x | x  Odd  x > 6 }
Datatype Declarations
• Examples
– data
Color = Red | Yellow | Blue
elements are Red, Yellow, Blue
data Atom = Atom String | Number Int
elements are Atom “A”, Atom “B”, …, Number 0, ...
data List
= Nil
|
Cons (Atom, List)
elements are Nil, Cons(Atom “A”, Nil), …
Cons(Number 2, Cons(Atom(“Bill”), Nil)), ...
• General form
– data <name> = <clause> | … | <clause>
<clause> ::= <constructor> | <contructor> <type>
– Type name and constructors must be Capitalized.
Datatypes and Pattern Matching
 Recursively defined data structure
data Tree = Leaf Int | Node (Int, Tree, Tree)
4
Node(4, Node(3, Leaf 1, Leaf 2),
Node(5, Leaf 6, Leaf 7))
3
1
sum (Leaf n) function
= n
 Recursive
5
2
sum (Node(n,t1,t2)) = n + sum(t1) + sum(t2)
6
7
Case Expression
 Datatype
data Exp = Var Int | Const Int | Plus (Exp, Exp)
 Case expression
case e of
Var n -> …
Const n -> …
Plus(e1,e2) -> …
Indentation matters in case statements in Haskell.
Evaluation by Cases
data Exp = Var Int | Const Int | Plus (Exp, Exp)
ev ( Var n) = Var n
ev ( Const n ) = Const n
ev ( Plus ( e1,e2 ) ) =
case ev e1 of
Var n -> Plus( Var n, ev e2)
Const n -> case ev e2 of
Var m -> Plus( Const n, Var m)
Const m -> Const (n+m)
Plus(e3,e4) -> Plus ( Const n,
Plus ( e3, e4 ))
Plus(e3, e4) -> Plus( Plus ( e3, e4 ), ev e2)
Laziness
 Haskell is a lazy language
 Functions and data constructors don’t evaluate
their arguments until they need them
cond :: Bool -> a -> a -> a
cond True t e = t
cond False t e = e
 Programmers can write control-flow operators
that have to be built-in in eager languages
Shortcircuiting
“or”
(||) :: Bool -> Bool -> Bool
True || x = True
False || x = x
Using Laziness
isSubString :: String -> String -> Bool
x `isSubString` s = or [ x `isPrefixOf` t
| t <- suffixes s ]
suffixes:: String -> [String]
type String = [Char]
-- All suffixes of s
suffixes[]
= [[]]
suffixes(x:xs) = (x:xs) : suffixes xs
or
-or
or
:: [Bool] -> Bool
(or bs) returns True if any of the bs is True
[]
= False
(b:bs) = b || or bs
A Lazy Paradigm
• Generate all solutions (an enormous tree)
• Walk the tree to find the solution you want
nextMove :: Board -> Move
nextMove b = selectMove allMoves
where
allMoves = allMovesFrom b
A gigantic (perhaps infinite)
tree of possible moves
Core Haskell
• Basic Types
–
–
–
–
–
–
–
–
Unit
Booleans
Integers
Strings
Reals
Tuples
Lists
Records
•
•
•
•
•
•
•
•
Patterns
Declarations
Functions
Polymorphism
Type declarations
Type Classes
Monads
Exceptions
Testing
• It’s good to write tests as you write code
• E.g. reverse undoes itself, etc.
reverse xs =
let rev ( [], z ) = z
rev ( y:ys, z ) = rev( ys, y:z )
in rev( xs, [] )
-- Write properties in Haskell
type TS = [Int]
-- Test at this type
prop_RevRev :: TS -> Bool
prop_RevRev ls = reverse (reverse ls) == ls
Test Interactively
Test.QuickCheck is
simply a Haskell library
(not a “tool”)
bash$ ghci intro.hs
Prelude> :m +Test.QuickCheck
Prelude Test.QuickCheck> quickCheck prop_RevRev
+++ OK, passed 100 tests
...with a strangelooking type
Prelude Test.QuickCheck> :t quickCheck
quickCheck :: Testable prop => prop -> IO ()
Demo QuickCheck
QuickCheck
• Generate random input based on type
– Generators for values of type a has type Gen a
– Have generators for many types
• Conditional properties
– Have form <condition> ==> <property>
– Example:
ordered xs = and (zipWith (<=) xs (drop 1 xs))
insert x xs = takeWhile (<x) xs++[x]++dropWhile (<x)
xs
prop_Insert x xs =
ordered xs ==> ordered (insert x xs)
where types = x::Int
QuickCheck
• QuickCheck output
– When property succeeds:
quickCheck prop_RevRev OK, passed 100 tests.
– When a property fails, QuickCheck displays a counter-example.
prop_RevId xs = reverse xs == xs where types = xs::[Int]
quickCheck prop_RevId
Falsifiable, after 1 tests: [-3,15]
• Conditional testing
– Discards test cases which do not satisfy the condition.
– Test case generation continues until
• 100 cases which do satisfy the condition have been found, or
• until an overall limit on the number of test cases is reached (to
avoid looping if the condition never holds).
See : http://www.cse.chalmers.se/~rjmh/QuickCheck/manual.html
Things to Notice
No side effects. At all.
reverse:: [w] -> [w]
 A call to reverse returns a new list; the old one
is unaffected.
prop_RevRev l = reverse(reverse l) == l
 A variable ‘l’ stands for an immutable value,
not for a location whose value can change.
 Laziness forces this purity.
Things to Notice
• Purity makes the interface explicit.
reverse:: [w] -> [w]
-- Haskell
• Takes a list, and returns a list; that’s all.
void reverse( list l )
/* C */
• Takes a list; may modify it; may modify
other persistent state; may do I/O.
Things to Notice
• Pure functions are easy to test.
prop_RevRev l = reverse(reverse l) == l
• In an imperative or OO language, you have to
– set up the state of the object and the external state it
reads or writes
– make the call
– inspect the state of the object and the external state
– perhaps copy part of the object or global state, so
that you can use it in the post condition
Things to Notice
Types are everywhere.
reverse:: [w] -> [w]
• Usual static-typing panegyric omitted...
• In Haskell, types express high-level design, in
the same way that UML diagrams do, with
the advantage that the type signatures are
machine-checked.
• Types are (almost always) optional: type
inference fills them in if you leave them out.
More Info: haskell.org
• The Haskell wikibook
– http://en.wikibooks.org/wiki/Haskell
• All the Haskell bloggers, sorted by topic
– http://haskell.org/haskellwiki/Blog_articles
• Collected research papers about Haskell
– http://haskell.org/haskellwiki/Research_papers
• Wiki articles, by category
•
– http://haskell.org/haskellwiki/Category:Haskell
Books and tutorials
– http://haskell.org/haskellwiki/Books_and_tutorials